Theorem curry1val 8087

Description: The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypothesis

Ref	Expression
curry1.1	⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))

Assertion

Ref	Expression
curry1val	⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = (𝐶𝐹𝐷))

Proof of Theorem curry1val

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		curry1.1	. . . 4 ⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))
2	1	curry1 8086	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))
3	2	fveq1d 6863	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷))
4		eqid 2730	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))
5	4	fvmptndm 7002	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅)
6	5	adantl 481	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅)
7		fndm 6624	. . . . . . 7 ⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
8	7	adantr 480	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → dom 𝐹 = (𝐴 × 𝐵))
9		simpr 484	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐷 ∈ 𝐵)
10	9	con3i 154	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐵 → ¬ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
11		ndmovg 7575	. . . . . 6 ⊢ ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐶𝐹𝐷) = ∅)
12	8, 10, 11	syl2an 596	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) = ∅)
13	6, 12	eqtr4d 2768	. . . 4 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
14	13	ex 412	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (¬ 𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷)))
15		oveq2 7398	. . . 4 ⊢ (𝑥 = 𝐷 → (𝐶𝐹𝑥) = (𝐶𝐹𝐷))
16		ovex 7423	. . . 4 ⊢ (𝐶𝐹𝐷) ∈ V
17	15, 4, 16	fvmpt 6971	. . 3 ⊢ (𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
18	14, 17	pm2.61d2 181	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
19	3, 18	eqtrd 2765	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = (𝐶𝐹𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ∅c0 4299  {csn 4592   ↦ cmpt 5191   × cxp 5639  ^◡ccnv 5640  dom cdm 5641   ↾ cres 5643   ∘ ccom 5645   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390  2^nd c2nd 7970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-1st 7971  df-2nd 7972

This theorem is referenced by:  nvinvfval  30576  hhssabloilem  31197